A219269 G.f. satisfies: A(x) = Sum_{n>=0} x^n/A(x)^n * Product_{k=0..n} k!.

1, 1, 1, 8, 247, 33184, 24678266, 125237615376, 5055581949347115, 1834887966372111613136, 6658588234946979374670842054, 265790194051800257952649093995518288, 127313960109916568757252293587045497552163302, 792786695940715289991550398242378268738388375150573312

Offset: 0

Author

Paul D. Hanna, Nov 16 2012

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + x^2 + 8*x^3 + 247*x^4 + 33184*x^5 + 24678266*x^6 +...
Given g.f. A(x), the table of coefficients in A(x)^n begins:
n=1: [(1), 1, 1, 8, 247, 33184, 24678266, 125237615376, ...];
n=2: [1,(2), 3, 18, 511, 66878, 49423458, 250524657604, ...];
n=3: [1, 3, (6), 31, 795, 101109, 74236366, 375861227934, ...];
n=4: [1, 4, 10,(48), 1103, 135908, 99117818, 501247428704, ...];
n=5: [1, 5, 15, 70,(1440), 171311, 124068685, 626683363390, ...];
n=6: [1, 6, 21, 98, 1812,(207360), 149089887, 752169136662, ...];
n=7: [1, 7, 28, 133, 2226, 244104,(174182400), 877704854447, ...];
n=8: [1, 8, 36, 176, 2690, 281600, 199347264,(1003290624000), ...]; ...
in which the main diagonal generates the superfactorials (A000178):
[1/1, 2/2, 6/3, 48/4, 1440/5, 207360/6, 174182400/7, 1003290624000/8, ...].
The logarithmic derivative of the superfactorials forms another diagonal:
A219266 = [1, 3, 31, 1103, 171311, 149089887, 877704854447, ...].
		

Crossrefs

Cf. A219266, A000178, A219270.

Programs

PARI

{a(n)=polcoeff(x/serreverse(x*sum(k=0,n+1,prod(j=0,k,j!)*x^k)+x^2*O(x^n)),n)}
for(n=0,21,print1(a(n),", "))

Formula

G.f. A(x) satisfies: [x^n] A(x)^(n+1)/(n+1) = Product_{k=0..n} k! = superfactorial A000178(n).

G.f.: x / Series_Reversion(x*F(x)) where F(x) = Sum_{n>=0} x^n*Product_{k=0..n} k! is the g.f. of A000178.